TITLE: How large is TREE(3)?
QUESTION [38 upvotes]: Friedman, in _Lectures notes on enormous integers shows that TREE(3) is much larger than n(4), itself bounded below by    $A^{A(187195)}(3)$ (where $A$ is the Ackerman function and exponentiation denotes iteration). But actually, using the fast-growing hierarchy, $n(p)$ is smaller than $f_{\omega^{\omega^\omega}}(p)$, shown by Friedman in

Long finite sequences, Journal of Combinatorial Theory, Series A 95 Issue 1 (2001) pp. 102-144, doi:10.1006/jcta.2000.3154, author pdf

while it seems that TREE grows faster than $f_{\Gamma_0}$ (${\Gamma_0}$ being the Feferman-Schütte ordinal). So it could well be that in fact TREE(3) is larger than, say, n(n(4)), or even any number expressible by iterations of n. What is known on this question?
For reference, I should have added that TREE(3) is the incredibly (at first, or even second look) large answer to the question : "which is the length of the  longest sequence $(T_2,T_3,T_4,\dots,T_n)$ of labeled trees such that $T_k$ has at most $k$ nodes labeled $a$ or $b$,   and $T_i$ is not a subtree of $T_j$ for $i < j$ ?".
Here, trees are rooted trees, and are treated as poset on their sets of vertices. A tree $T$ is called a subtree of $T'$ if there is an inf-preserving embedding from $T$ into $T'$, (that is, an injective map $h:Vertices(T) \to Vertices(T')$ such that $h(\inf(x,y)) = \inf((h(x), h(y))$) that respects the labeling by $a$ or $b$.

REPLY [2 votes]: I'd like to post an improvement of what originally posted here, by Deedlit and myself. Another related post is here, which shows an even better bound, though there are questions how the order in the last post exaclty works, i.e. it does not contain evidence why it should be like this. Anyway, we start with these trees:
T1  {}
T2  [[]]
T3  [()()]
T4  [(()())]
T5  [(((())))]
T6  ([((()))][])
T7  ([((()))]()())
T8  ([((()))]()())
T9  ([((()))]((())))
T10 ([((()))](()()))
T11 ([((()))](((()))))
T12 ([((()))]((())))
T13 ([((()))](()))
T14 ([((()))]())
T15 ([((()))])
T16 ([(())][(())][(())][(())][(())])
T17 ([(())][(())][(())][(())]([()][]))
T18 ([(())][(())][(())][(())]([()][]))
T19 ([(())][(())][(())][(())]([()]()()))
T20 ([(())][(())][(())][(())]([()]())[()])
T21 ([(())][(())][(())][(())]([()]())(([])))
T22 ([(())][(())][(())][(())]([()]())([][][]))
T23 ([(())][(())][(())][(())]([()]())([][])([]))
T24 ([(())][(())][(())][(())]([()]())([][])[][][])
T25 ([(())][(())][(())][(())]([()]())([][])[][]()())
T26 ([(())][(())][(())][(())]([()]())([][])[][]((())))
T27 ([(())][(())][(())][(())]([()]())([][])[][](()))
T28 ([(())][(())][(())][(())]([()]())([][])[][]())
T29 ([(())][(())][(())][(())]([()]())([][])[][])
T30 ([(())][(())][(())][(())]([()]())([][])[]#)

where # is the first tree of $\mathrm{tree}(8)$. 
This is very important. We can now reduce # for $\mathrm{tree}(8)$ steps. In general, we can reduce [] for as many steps as it allows. Generally this will be comparable to the number of brackets allowed in total. 
This means that we can preform basic recursion around tree already. We can spawn a ([]) to $n$ []s, so this is already at $f_{1}$ where $f_0 = \mathrm{tree}$.  We can spawn a () to $n$ ([])s so it is at at $f_2$. Similiarly, we can take ([]#) for any tree # and reduce it just like as we would as in tree. We can reach $f_{\vartheta(\Omega^{\omega})}(n)$ in the modified hierarchy this way. $f$ is a certain variant of the fast growing hierarchy. 
But $f_{\vartheta(\Omega^{\omega})}(n)$ in the modified hierarchy is just $F_{\vartheta(\Omega^{\omega},0)2}(n)$, where $F$ is a certain variant of the normal fast growing hierarchy (with $F_0=n+1$.)
Similiarly, we have ([][]#), reaching $F_{\vartheta(\Omega^{\omega})3}(n)$ at ([][][]) and $F_{\vartheta(\Omega^{\omega})\omega}(n)$ at (([])).  $F_{\vartheta(\Omega^{\omega})(\omega+1)}(n)$ is reached by  (([])[]), $F_{\vartheta(\Omega^{\omega})(\omega2)}(n)$ by  (([])([])), and $F_{\vartheta(\Omega^{\omega})\omega^2}(n)$ at (()).  With # we work in the product, reaching $F_{\vartheta(\Omega^{\omega})^2}(n)$ at (([][])) and $F_{\vartheta(\Omega^{\omega})^\omega}(n)$ at ((([]))).
A pattern emerges. We can in fact reach $F_{\varepsilon_{\vartheta(\Omega^{\omega})+1}}$ at [()] which would evaluate to (((...((([])))...))) with as many pairs as possible. 
We see that TREE(3) is already far beyond the tree function itself.